Question

Question

Which of these graphs shows that the linear system 3x+2y=12
and 6x+4y=24
has an infinite number of solutions?(1 point)
Responses

A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. Two intersecting lines are plotted on the plane. A solid downward slanting line passes through the points left parenthesis negative 6 comma 3 right parenthesis and left parenthesis negative 4 comma 0 right parenthesis. A dotted upward slanting line passes through the points left parenthesis 0 comma negative 6 right parenthesis and left parenthesis 6 comma 4 right parenthesis.
Image with alt text: A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. Two intersecting lines are plotted on the plane. A solid downward slanting line passes through the points left parenthesis negative 6 comma 3 right parenthesis and left parenthesis negative 4 comma 0 right parenthesis. A dotted upward slanting line passes through the points left parenthesis 0 comma negative 6 right parenthesis and left parenthesis 6 comma 4 right parenthesis.

A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. A line is plotted on the plane. A downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 4 comma 0 right parenthesis.
Image with alt text: A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. A line is plotted on the plane. A downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 4 comma 0 right parenthesis.

A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. Two parallel lines are plotted on the plane. A dotted downward slanting line with arrows at both ends passes through the points left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma negative 6 right parenthesis. A solid downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 4 comma 0 right parenthesis.
Image with alt text: A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. Two parallel lines are plotted on the plane. A dotted downward slanting line with arrows at both ends passes through the points left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma negative 6 right parenthesis. A solid downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 4 comma 0 right parenthesis.

A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. A line is plotted on the plane. A downward slanting line with arrows at both ends passes through the points left parenthesis negative 10 comma 9 right parenthesis and left parenthesis negative 4 comma 0 right parenthesis.
Image with alt text: A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. A line is plotted on the plane. A downward slanting line with arrows at both ends passes through the points left parenthesis negative 10 comma 9 right parenthesis and left parenthesis negative 4 comma 0 right parenthesis.
Skip to navigation

Answer 1

To determine which of the graphs shows that the linear system \(3x + 2y = 12\) and \(6x + 4y = 24\) has an infinite number of solutions, we first need to analyze the equations.

Notice that the second equation, \(6x + 4y = 24\), can be simplified by dividing everything by 2:

\[ 3x + 2y = 12 \]

This is exactly the same as the first equation. Therefore, both equations represent the same line. When two equations represent the same line, they share all points along that line, resulting in an infinite number of solutions.

In graphical terms, this situation is represented by two overlapping lines.

From the response options you provided, the graph that illustrates this relationship will show one line completely overlapping another line.

• If you see two lines that are parallel, or completely different (intersecting at one point), those scenarios indicate either no solutions or a unique solution.
• But if the graph shows two lines that lie on top of each other, it confirms that there are infinite solutions.

Thus, you should look for the option that depicts two identical lines (one solid, perhaps, and the other dotted, both fully overlapping) to identify the scenario where the linear system has an infinite number of solutions.